Finding the equation of a straight line

Finding the equation of a straight line from two points

Let’s suppose we know that two points are on a straight line - (x, y) and (x_1, y_1) - and we want to find the equation of that straight line.

We need to know the gradient if we want to find the equation of the line, and after that, we can substitute in our values for x, y, x_1, y_1 and m into the equation of a straight line: y - y_1 = m(x - x_1).

Finding the gradient

The gradient of a graph can be found by calculating the change in y divided by the change in x, or:

m = \frac{\Delta y}{\Delta x}

• The change in y is going to be the difference between the starting y value - y_1 - and the ending y value - y, so \Delta y = y - y_1.
• The change in x is going to be the difference between the starting x value - x_1 - and the ending x value - x, so \Delta x = x - x_1.

So, the gradient can be calculated using:

m = \frac{y - y_1}{x - x_1}

Finding the equation of the line

Once you have found the gradient, you can substitute it into the equation of a straight line:

y - y_1 = m(x - x_1)

Example: find the equation of the straight line that passes through the points (2, 3) and (5, 11)

• Find the gradient:
  • m = \frac{y - y_1}{x - x_1}
  • = \frac{11 - 3}{5 - 2}
  • = \frac{8}{3}
• Substitute into the equation of a straight line:
  • y - y_1 = m(x - x_1)
  • y - 3 = \frac{8}{3}(x - 2)
• We can rearrange this into the form y = mx + c if we want to:
  • y - 3 = \frac{8}{3}x - \frac{16}{3}
  • y = \frac{8}{3}x - \frac{16}{3} + 3
  • y = \frac{8}{3}x - \frac{16}{3} + \frac{9}{3}
  • y = \frac{8}{3}x - \frac{7}{3}
• Answer: y = \frac{8}{3}x - \frac{7}{3}

Example: find the equation of the straight line that passes through the points (1, 4) and (3, 10)

• Find the gradient:
  • m = \frac{y - y_1}{x - x_1}
  • = \frac{10 - 4}{3 - 1}
  • = \frac{6}{2}
  • = 3
• Substitute into the equation of a straight line:
  • y - y_1 = m(x - x_1)
  • y - 4 = 3(x - 1)
• We can rearrange this into the form y = mx + c if we want to:
  • y - 4 = 3x - 3
  • y = 3x - 3 + 4
  • y = 3x + 1
• Answer: y = 3x + 1

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